I have a problem where there are two agents in a joint project, Each agent $i$ puts in effort $x_i$ $(0 \leq x_i \leq 1)$ which cost each $c(x_i)= x_i^2$. The outcome of the project is $$ f(x_1, x_2)= 3 x_1 x_2 $$
which is split equally between both, regardless of their effort levels. I am asked to formulate de situation as a normal form game and find the NE.
What I have done is that I have assumed some values for the efforts to evaluate the game and I have also maximized each player's payoff individually to get best responses and make them intersect to get to the NE.
$\max f(x_1, x_2)-c_1$ with respect to $x_1$ ; yields : $x_1 = (3/4)x_2$ and $\max f(x_1, x_2)-c_2$ with respect to $x_2$ ; yields : $x2 = (3/4)x_1$ Both maximization problems intersect when $x_1 = 0 = x_2$
Anyhow, I don't think this is the right way to go...